A fast new algorithm for weak graph regularity

TL;DR

This paper introduces a deterministic, efficient algorithm for approximating graph structures using regular partitions and applies it to estimate subgraph counts with high accuracy.

Contribution

It presents a new deterministic algorithm for finding epsilon-regular partitions of graphs in near-quadratic time, improving the efficiency of graph regularity methods.

Findings

Algorithm runs in $ ext{ extasciicircum}O(1)$-powered polynomial time

Provides accurate estimates of subgraph counts within additive error

Enables deterministic approximation of graph regularity and subgraph enumeration

Abstract

We provide a deterministic algorithm that finds, in $\epsilon^{-O(1)} n^2$ time, an $\epsilon$-regular Frieze-Kannan partition of a graph on $n$ vertices. The algorithm outputs an approximation of a given graph as a weighted sum of $\epsilon^{-O(1)}$ many complete bipartite graphs. As a corollary, we give a deterministic algorithm for estimating the number of copies of $H$ in an $n$-vertex graph $G$ up to an additive error of at most $\epsilon n^{v(H)}$, in time $\epsilon^{-O_H(1)}n^2$.